In this setting it is probably useful to point out that the
preservation or failure of the “duality” or “complementarity”
between \({\cal H}_{math}\)  and \({\cal H}_{phys}\)  may be fairly
sensitive to the detailed properties of the Hamiltonian in question.
For a word of warning one does not even need to go to the
non-Hermitian quantum field theory where the Stone theorem need not
apply. Indeed, it is fully sufficient to see that multiple subtle
formal problems like, e.g., the possibility of the non-existence of
the metric may emerge in a finite-dimensional model (see, e.g.,
section Nr. 3 in review [1]}).
